2. Decimal number system
• The decimal system contains ten unique symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
• Its base or radix is 10.
• The leftmost digit in any number representation, which has the greatest
positional weight out of all the digits present in that number, is called the most
significant digit (MSD).
• The rightmost digit in any number representation, which has the least positional
weight out of all the digits present in that number, is called the least significant
digit (LSD).
• Eg. Consider the decimal number 9256.26 is represented as:
9256.26 = 9x1000+2x100+5x10+6x1+2x(1/10)+6x(1/100)
= 9x103+2x102+5x101+6x100+2x10−1+6x10−2
3. • Subtraction of decimal numbers can be accomplished by the 9’s and
10’s complement methods.
• Similar to the 1’s and 2’s complement methods of binary.
• The 9’s complement of a decimal number is obtained by subtracting
each digit of that decimal number from 9.
• The 10’s complement of a decimal number is obtained by adding a 1
to its 9’s complement.
8. Binary to decimal conversion
• Convert 10101 base 2 (101012) to decimal.
• Convert 11011.101 base 2 (11011.1012) to decimal.
• Convert 1001011 base 2 (10010112)to decimal
10. Decimal to binary conversion
• Convert (163.87510) to binary.
• Convert (5210) to binary.
• Convert (0.7510) to binary.
• Convert (105.1510) to binary.
30. What is Restoring method?
• The hardware method just described is called the RESTORING METHOD. The
reason for the name is that the partial remainder is restored by adding the divisor
to the negative difference. Two other methods are available for dividing numbers,
the COMPARISION method and the NONRESTORING method. In the
comparison method A and B are compared prior to the subtraction operation .
• No restoring method B is not added if the difference is negative but instead, the
negative difference is shifted left and then B is added.
42. Floating Point Representation
• It has mainly three parts:
1. Mantissa
2. Base
3. Exponent
Number Mantissa Base Exponent
3x106 3 10 6
110x28 110 2 8
6132.784 6132784 10 -3
43. IEEE754 floating point number representation
A) Single precision format: (32 bits)
1 bit 8bits 23 bits
B) Double precision format: (64 bits)
1 bit 11 bits 52bits
Sign Exponent Mantissa
Sign Exponent Mantissa
44. E.g. Represent (1259.125) base 10 in single
and double precision format.
• Step 1: convert decimal number to binary number.
(1259.125)10 = (10011101011.001)2
• Step 2: Normalize the number.
For single precision = (1.N) 2𝐸−127
For double precision = (1.N) 2𝐸−1023
10011101011.001 = 1. 0011101011001 x 210
45. • Step 3: Single precision format.
For single precision = (1.N) 2𝐸−127= = 1. 0011101011001 x 210
𝐸 − 127 = 10; 𝐸 = 137. (𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑖𝑛 8 𝑏𝑖𝑡)
Mantissa= 0011101011001
Base= 2
Exponent = 137 in 8 bits = 10001001 base 2
Sign = positive (0)
1 bit 8bits 23bits
0 10001001 0011101011001…..000….
